Solution 15.

See text and/or instructor's solution manual.

Solution.   Use the fact that

           

Now expand in a Taylor series about      

          

Then apply Corollary 7.2 in a manner similar to Example 7.2 and integrate the series term by term along the line segment from  .

              
            
            and
            
            

So that    which can be rearranged to get

            ,

and the series converges for    or  .  

We are done.   

                    The images of the disk    under   ,  for  .  

                    The image of the disk    under the mapping   .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell